Storage only: Amazon S3, Crashplan, Mozy, Backblaze. These solutions are backup/storage services, where you can view the files online, but not photos. In otherwords, you can confirm a list of image files uploaded, but not view the images themselves. Amazon S3 has APIs, all others use their own client interfaces. These have the benefit of being able to store not only images, but also any other file type, including LR or Aperture library files, psd, tiff, dng files.

Viewing and sharing. Thousands of options again. Services such as Flickr, Picasa, 500px, focus on sharing of JPEGs only, though of course are a sort of 'backup', since these files live in the 'cloud'. Typically, bulk download can be much more troublesome than solutions in #2 above.

This is a good, helpful answer. Would you consider moving it to the other question this is a duplicate of?
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mattdmApr 5 '13 at 17:32

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Another to add to the list is Secure Capsule, it's an archiving service backed by AWS S3 and glacier, you get a heap of free storage, the large plans are really affordable and we offer immediate access to thumbnails and previews. The raw files can be restored after about 4 hours (they come off glacier) which keeps the cost really low. Disclaimer I am the founder of Secure Capsule.
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rabsApr 6 '13 at 1:17